Re: Cantor Diagonal Proof

On 08/04/2025 19:44, Andy Walker wrote:

On 08/04/2025 16:17, Richard Heathfield wrote:

It will, however, take me some extraordinarily convincing
mathematics before I'll be ready to accept that 1/3 is irrational.

      I don't think that's quite what Wij is claiming.  He thinks,
rather, that 0.333... is different from 1/3.  No matter how far you
pursue that sequence, you have a number that is slightly less than
1/3.

And Achilles never /quite/ catches the tortoise...

In real analysis, the limit is 1/3 exactly.  In Wij-analysis, limits don't exist [as I understand it], because he doesn't
accept that there are no infinitesimals.  It's like those who
dispute that 0.999... == 1 [exactly], and when challenged to
produce a number between 0.999... and 1, produce 0.999...5.

0.999...5 isn't equal to 1, but neither is it a recurring decimal. They might as well claim that 0.95 isn't 1. Well, of course it isn't. But neither is it 0.9r.
MathWorld defines infinitesimals as follows: "An infinitesimal is some quantity that is explicitly nonzero and yet smaller in absolute value than any real quantity."
I don't plan to dwell on this, but I'm tempted to wonder how infinitesimals fare under division.

They have a point, as the Archimedean axiom is not one of the
things that gets mentioned much at school or in many undergrad
courses, and it seems like an arbitrary and unnecessary
addition to the rules.  But we have no good and widely-known
notation for what can follow a "..."

https://en.wikipedia.org/wiki/Racing_flags#Chequered_flag
...because Achilles /does/ win the race... at the limit.

so the Wijs of this world get mocked.

I will forbear.
--
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
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